A022004 -id:A022004 - cp's OEIS frontend

A236464 Primes p with prime(p) + 2 and prime(p) + 6 both prime.

3, 5, 7, 13, 43, 89, 313, 613, 643, 743, 1171, 1279, 1627, 1823, 1867, 1999, 2311, 2393, 2683, 2753, 2789, 3571, 4441, 4561, 5039, 5231, 5647, 5953, 6067, 6317, 6899, 8039, 8087, 8753, 8923, 9337, 9787, 9931, 10259, 10667

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236472, this sequence contains infinitely many terms, i.e., there are infinitely many prime triples of the form {prime(p), prime(p) + 2, prime(p) + 6} with p prime.

See A236462 for a similar sequence.

			a(1) = 3 since 3, prime(3) + 2 = 7 and prime(3) + 6 = 11 are all prime, but prime(2) + 6 = 9 is composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A022004, A236457, A236458, A236462, A236472.

p[n_]:=p[n]=PrimeQ[Prime[n]+2]&&PrimeQ[Prime[n]+6]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]

A237769 Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 3, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 1, 1, 2, 2, 3, 4, 3, 3, 4, 3, 5, 5, 3, 3, 2, 2, 5, 5, 3, 3, 3, 3, 5, 5, 2, 2, 3, 3, 3, 4, 2, 2, 6, 6, 9, 8, 4, 4, 3, 3, 6, 6, 5, 5, 4, 4, 7

Offset: 1

Zhi-Wei Sun, Feb 13 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 8, and a(n) = 1 only for n = 9, 34, 35.
(ii) For any integer n > 4, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) + 5 are all prime. Also, for each integer n > 8, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) - 5 are all prime.
(iii) For any integer n > 6, there is a prime p < n such that phi(n-p) - 1 and phi(n-p) + 1 are both prime, where phi(.) is Euler's totient function.

			a(9) = 1 since 2, pi(9-2) - 1 = 3 and pi(9-2) + 1 = 5 are all prime.
a(34) = 1 since 19, pi(34-19) - 1 = pi(15) - 1 = 5 and pi(34-19) + 1 = pi(15) + 1 = 7 are all prime.
a(35) = 1 since 19, pi(35-19) - 1 = pi(16) - 1 = 5 and pi(35-19) + 1 = pi(16) + 1 = 7 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000720, A001359, A006512, A014574, A022004, A022005, A237705, A237706, A237768.

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
a[n_]:=Sum[If[TQ[PrimePi[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A078561 p, p+4 and p+10 are consecutive primes.

Original entry on oeis.org

19, 43, 79, 127, 163, 229, 349, 379, 439, 499, 643, 673, 937, 967, 1009, 1093, 1213, 1279, 1429, 1489, 1549, 1597, 1609, 2203, 2347, 2389, 2437, 2539, 2689, 2833, 2953, 3079, 3319, 3529, 3613, 3793, 3907, 3919, 4003, 4129, 4447, 4639, 4789, 4933, 4999

Offset: 1

Labos Elemer, Dec 10 2002

nonn

Subsequence of A029710. - R. J. Mathar, May 06 2017

			Between p and p+10 [46] difference-pattern: 19(4)23(6)29;

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous inter-prime d-patterns with d<=6: A022004[24], A022005[42], A049437[26], A049438[62], A078561[46], A078562[64], A047948[66].

Select[Prime@ Range[10^3], Differences@ NestList[NextPrime, #, 2] == {4, 6} &] (* Michael De Vlieger, May 06 2017 *)
Select[Partition[Prime[Range[700]],3,1],Differences[#]=={4,6}&][[All,1]] (* Harvey P. Dale, Mar 24 2018 *)

isok(p) = isprime(p) && (nextprime(p+1) == p+4) && (nextprime(p+5) == p+10); \\ Michel Marcus, Dec 20 2013

is(n)=isprime(n) && isprime(n+4) && isprime(n+10) && !isprime(n+6) && n>3 \\ Charles R Greathouse IV, Dec 20 2013

A078562 p, p+6 and p+10 are consecutive primes.

Original entry on oeis.org

31, 61, 73, 157, 271, 373, 433, 607, 733, 751, 1291, 1543, 1657, 1777, 1861, 1987, 2131, 2287, 2341, 2371, 2383, 2467, 2677, 2791, 2851, 3181, 3313, 3607, 3691, 4441, 4507, 4723, 4993, 5407, 5431, 5521, 5563, 5641, 5683, 5851, 6037, 6211, 6571, 6961

Offset: 1

Labos Elemer, Dec 10 2002

nonn

Subsequence of A031924. - R. J. Mathar, Jun 15 2013

			Between p and p+10 the difference-pattern is [64] like e.g. for p=31: 31(6)37(4)41.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous inter-prime d-patterns with d<=6: A022004[24], A022005[42], A049437[26], A049438[62], A078561[46], A078562[64], A047948[66].

Transpose[Select[Partition[Prime[Range[1000]],3,1],#[[3]]-#[[1]]==10&&#[[2]]-#[[1]]==6&]][[1]] (* Harvey P. Dale, Dec 09 2010 *)

A186634 Irregular triangle, read by rows, giving dense patterns of n primes.

Original entry on oeis.org

0, 2, 0, 2, 6, 0, 4, 6, 0, 2, 6, 8, 0, 2, 6, 8, 12, 0, 4, 6, 10, 12, 0, 4, 6, 10, 12, 16, 0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20, 0, 2, 6, 8, 12, 18, 20, 26, 0, 2, 6, 12, 14, 20, 24, 26, 0, 6, 8, 14, 18, 20, 24, 26, 0, 2, 6, 8, 12, 18, 20, 26, 30, 0, 2, 6, 12, 14, 20, 24, 26, 30, 0, 4, 6, 10, 16, 18, 24, 28, 30, 0, 4, 10, 12, 18, 22, 24, 28, 30, 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 0, 2, 6, 12, 14, 20, 24, 26, 30, 32

Offset: 2

T. D. Noe, Feb 24 2011

The first pattern (0,2) is for twin primes (p,p+2). Row n contains A083409(n) patterns, each one consisting of 0 followed by n-1 terms. In each row the patterns are in lexicographic order.

These numbers (in a slightly different order) appear in Table 1 of the paper by Tony Forbes. Sequence A186702 gives the least prime starting a given pattern.

			The irregular triangle begins:
0, 2
0, 2, 6, 0, 4, 6
0, 2, 6, 8
0, 2, 6, 8, 12, 0, 4, 6, 10, 12
0, 4, 6, 10, 12, 16
0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20

T. D. Noe, Rows n = 2..20, flattened (from Forbes)
Thomas J. Engelsma, Permissible Patterns
Tony Forbes, Prime clusters and Cunningham chains, Math. Comp. 68 (1999), 1739-1747.
Tony Forbes, Smallest Prime k-tuples
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A020497, A008407, A186702, and sequences created by these patterns: A001359, A022004, A022005, A007530, A022006, A022007, A022008, A022009, A022010, A022011, A022012, A022013, A022545, A022546, A022547, A022548, A027569, A027570.

A372209 Primes p_1 where products m of k = 3 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

Original entry on oeis.org

3, 7, 13, 23, 31, 37, 47, 53, 61, 67, 73, 89, 97, 103, 113, 131, 139, 151, 157, 167, 173, 181, 193, 199, 211, 223, 233, 241, 251, 257, 263, 271, 277, 293, 307, 317, 337, 359, 367, 373, 389, 409, 421, 433, 449, 457, 467, 479, 491, 509, 523, 547, 557, 563, 577, 587

Offset: 1

Michael De Vlieger, Sep 11 2024

Primes p such that the second differences of p and the next 2 primes is never positive.
Superset of A022005.
Does not intersect A022004.

			3 is in the sequence since m = 3*5*7 = 105 is such that 3 is less than the cube root of 105, but both 5 and 7 exceed it.
5 is not in the sequence because m = 5*7*11 = 385 is such that both 5 and 7 are less than the cube root.
7 is in the sequence since m = 7*11*13 = 1001 is such that 7 < 1001^(1/3), but both 11 and 13 are larger than 1001^(1/3), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022004, A022005, A375008.

k = 3; s = {1}~Join~Prime[Range[k - 1]]; Reap[Do[s = Append[Rest[s], Prime[i + k - 1]]; r = Surd[Times @@ s, k]; If[Count[s, _?(# < r &)] == 1, Sow[Prime[i]] ], {i, 600}] ][[-1, 1]]

A162001 Initial members of prime triples (p, p+2, p+6) for which also the sum 3p+8 is prime.

Original entry on oeis.org

5, 11, 17, 41, 101, 311, 347, 641, 857, 1301, 1427, 1481, 2237, 2687, 3461, 3527, 4001, 4787, 8861, 10457, 11171, 11777, 13691, 14627, 19421, 19991, 21017, 21557, 22271, 24917, 25997, 26261, 26681, 26711, 27737, 29021, 31511, 32057, 33347, 35591

Offset: 1

Milton L. Brown (miltbrown(AT)earthlink.net), Jun 24 2009

nonn

A subsequence of A022004 (= initial members of prime triples (p, p+2, p+6)). - Emeric Deutsch, Jul 12 2009

			(5,7,11) => 23 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A162001.

Cf. A022004, A376013.

a := proc (n) if isprime(n) = true and isprime(n+2) = true and isprime(n+6) = true and isprime(3*n+8) = true then n else end if end proc: seq(a(n), n = 1 .. 50000); # Emeric Deutsch, Jul 12 2009

Select[Select[Partition[Prime[Range[4000]], 3, 1], Differences[#] == {2, 4} &], PrimeQ[Total[#]] &][[;; , 1]] (* Amiram Eldar, Sep 06 2024 *)

list(lim)=my(v=List(), p=5, q=7, s); forprime(r=11, lim+6, if(r-p==6 && q-p==2 && isprime(s=3*p+8), listput(v, p)); p=q; q=r); Vec(v) \\ Charles R Greathouse IV, Sep 19 2024

a(n) == 5 (mod 6). - Hugo Pfoertner, Sep 06 2024
a(n) = (A376013(n) - 8)/3. - Amiram Eldar, Sep 06 2024
a(n) >> n log^4 n. - Charles R Greathouse IV, Sep 19 2024

Definition corrected by Emeric Deutsch, Jul 12 2009

Extended by Emeric Deutsch, Jul 12 2009

A181994 Initial members of prime triples p < q < r such that r-q = n*(q-p).

Original entry on oeis.org

3, 2, 29, 8117, 137, 197, 45433, 1931, 521, 156151, 1949, 1667, 480203, 2969, 7757, 2181731, 12161, 28349, 6012893, 20807, 16139, 3933593, 163061, 86627, 13626251, 25469, 40637, 60487753, 79697, 149627, 217795241, 625697, 552401, 240485251, 173357, 360089, 122164741

Offset: 1

Zak Seidov, May 31 2012

nonn

For some n, a(n) are abnormally large: note, e.g., that if q-p=2, then n cannot be of the form 4+3k, that is why a(4), a(7), a(10), ... are larger than neighbor terms; also, a(67) > 1.1*10^11. Is the sequence infinite?

			First 10 cases of {n,p,q,r}: {1,3,5,7}, {2,2,3,5}, {3,29,31,37}, {4,8117,8123,8147}, {5,137,139,149}, {6,197,199,211}, {7,45433,45439,45481}, {8,1931,1933,1949}, {9,521,523,541}, {10,156151,156157,156217}.

Zak Seidov, Table of n, a(n) for n = 1..66

Particular cases with q-p=2: A022004 [(r-q)=2*(q-p)], A049437 [(r-q)=3*(q-p) starting with 2nd term].

Cf. A179210.

a(n) = prevprime(A179210(n)). - Robert G. Wilson v, Dec 23 2016

A201599 Initial primes in prime triples (p, p+2, p+6) preceding the maximal gaps in A201598.

Original entry on oeis.org

5, 17, 41, 107, 347, 461, 881, 1607, 2267, 2687, 6197, 6827, 39227, 46181, 56891, 83267, 167621, 375251, 381527, 549161, 741677, 805031, 931571, 2095361, 2428451, 4769111, 4938287, 12300641, 12652457, 13430171, 14094797, 18074027, 29480651, 107379731, 138778301, 156377861

Offset: 1

Alexei Kourbatov, Dec 03 2011

nonn

Prime triples (p, p+2, p+6) are one of the two types of densest permissible constellations of 3 primes. Maximal gaps between triples of this type are listed in A201598; see more comments there.

			The gap of 6 between triples starting at p=5 and p=11 is the very first gap, so a(1)=5. The gap of 6 between triples starting at p=11 and p=17 is not a record, so it does not contribute to the sequence. The gap of 24 between triples starting at p=17 and p=41 is a maximal gap - larger than any preceding gap; therefore a(2)=17.

Alexei Kourbatov, Table of n, a(n) for n = 1..72
Tony Forbes, Prime k-tuplets
Alexei Kourbatov, Maximal gaps between prime triples
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022004 (prime triples p, p+2, p+6), A201598.

A236472 a(n) = |{0 < k < n: p = prime(k) + phi(n-k), prime(p) + 2 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 2, 3, 0, 1, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 2, 2, 1, 0, 0, 3, 1, 2, 0, 2, 2, 2, 1, 0, 0, 4, 1, 0, 0, 0, 0, 5, 0, 1, 1, 1, 2, 1, 1, 3, 0, 0, 2, 2, 0, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 2, 0, 0, 2, 1, 1, 3, 0, 0, 2, 0, 3, 0, 0, 1, 1, 0, 2, 0, 0

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: a(n) > 0 for every n = 330, 331, ....
We have verified this for n up to 80000.
The conjecture implies that there are infinitely many prime triples of the form {prime(p), prime(p) + 2, prime(p) + 6} with p prime. See A236464 for such primes p.

			a(10) = 1 since prime(2) + phi(8) = 3 + 4 = 7, prime(7) + 2 = 17 + 2 = 19 and prime(7) + 6 = 23 are all prime.
a(877) = 1 since prime(784) + phi(877-784) = 6007 + 60 = 6067, prime(6067) + 2 = 60101 + 2 = 60103 and prime(6067) + 6 = 60107 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A022004, A236456, A236460, A236464, A236467, A236470.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[n]+6]
f[n_,k_]:=Prime[k]+EulerPhi[n-k]
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A236464 Primes p with prime(p) + 2 and prime(p) + 6 both prime.

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A237769 Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.

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A078561 p, p+4 and p+10 are consecutive primes.

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A078562 p, p+6 and p+10 are consecutive primes.

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A186634 Irregular triangle, read by rows, giving dense patterns of n primes.

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A372209 Primes p_1 where products m of k = 3 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

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A162001 Initial members of prime triples (p, p+2, p+6) for which also the sum 3p+8 is prime.

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A181994 Initial members of prime triples p < q < r such that r-q = n*(q-p).

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